All sanctions are sought for justice, not revenge.
As an accepted fact, the death penalty has greater due process protections for the defendant/convicted murderer than does any other sanctions, removing the death penalty even further from revenge than all other sanctions.
Due process removed revenge from the justice equation.
No one connected to the crime can be a fact finder in any case, nor can they determine the verdict nor the sentence at trial, all of which are the province of the independent fact finders, those being either judge or jury, neither of which have a motive for revenge.
This is, all, very well known.
If an actual innocent is convicted, it is done, solely, based upon neither the judge nor jury being aware of the defendant's actual innocence - to the contrary, the evidence provided proved the defendant guilty, beyond a reasonable doubt. The judge or jury found that justice was served by the verdict.
Unknown to judge or jury, the verdict is unjust.
Thereafter, it is the province of the appellate courts and/or the executive branch and/or post conviction provisions in law, providing for actual innocence claims to be heard, to find and determine legal error and/or actual innocence and/or a level of uncertainty in the verdict so that the unjust verdict is reversed and justice found.
Also very well known.